It takes about 45 minutes for any object to pass entirely through the earth, at abotu any angle, as long as it is not affected by forces other than gravity (and the mass is negligible...but if its not, that cna be added in quite simply).

You should also add that the earth is not spinning in your setup. If it were spinning and an object were to fall in at a vector perpendicular to the earth's axis, they would bounce back and forth on the walls.

If you need to be concerned with the velocity at the center of the earth, jsut use conservation of energy:

[tex]E = .5mv^2 - \frac{GMm}{r}[/tex]

where [tex]M[/tex] is the mass that's under the falling object, in other words, if at some arbitrary point inside the earth, your distance from the center is R, then R is the radius of the sphere you need to worry about. The earth that you've passed through doesnt matter.